URL:http://www.emath.fr/cocv/ DOI: 10.1051/cocv:2002018
SOME REGULARITY RESULTS FOR MINIMAL CRYSTALS
L. Ambrosio
1, M. Novaga
2and E. Paolini
1Abstract. We introduce an intrinsic notion of perimeter for subsets of a general Minkowski space (i.e.
a finite dimensional Banach space in which the norm is not required to be even). We prove that this notion of perimeter is equivalent to the usual definition of surface energy for crystals and we study the regularity properties of the minimizers and the quasi-minimizers of perimeter. In the two-dimensional case we obtain optimal regularity results: apart from a singular set (which is H1-negligible and is empty when the unit ball is neither a triangle nor a quadrilateral), we find that quasi-minimizers can be locally parameterized by means of a bi-lipschitz curve, while sets with prescribed bounded curvature are, locally, lipschitz graphs.
Mathematics Subject Classification. 49J45, 49Q20. Received October 4, 2001.
1. Introduction
In this paper we consider the minimizers and the quasi-minimizers of the functional
E7→ P (E) :=
Z
∂∗Eϕ(νE(x)) dH
n−1(x), (1)
defined for a set E⊂ Rnwith finite perimeter. In (1) ∂∗E represents the reduced boundary of the set E (i.e. the
points of the boundary where a generalized normal vector is defined), νE(x) denotes the exterior unit normal vector to E at x ∈ ∂∗E, and ϕ:Rn → [0, ∞[ is a positively 1-homogeneous convex function. We observe that,
when the function ϕ is even (hence 1-homogeneous), the functional (1) provides an intrinsic notion of perimeter when one endowesRn with a suitable Banach structure related to ϕ (see Th. 2.7).
The minima of (1) have been widely studied in the literature [3, 9, 26], in particular it is well known [3, 10] that, whenever ϕ is smooth and uniformly elliptic out of the origin, the minima are hyper-surfaces of class
C1,α out of a “small” singular set. Here, we are mainly interested in the case of general convex functions ϕ.
In this situation it is quite easy to provide examples of minima which are locally the graph of a lipschitz, but not differentiable, function. Moreover, the boundary of these sets may have singular points (i.e. points where
∂E is not a manifold) also in two dimensions. These examples have been studied in particular by Taylor and
Cahn [29], Morgan [23]), which have also classified the singular cones which are minimal for the functional
Keywords and phrases: Quasi-minimal sets, Wulff shape, crystalline norm.
1Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy; e-mail: ambrosio@emath.ethz.ch 2Dipartimento di Matematica, Universit`a di Pisa, via F. Buonarroti 2, 56126 Pisa, Italy.
c
in (1). As has been pointed out by Taylor in [26], the study of such minimal sets is closely related to problems arising in material science and phase transitions in an anisotropic environment.
We introduce a class of sets which are ω-minimal for (1) (see Def. 3.1), including the minima for
E7→ P (E) +
Z
EH(x) dx, (2)
with H ∈ Lp(Rn), p≥ n. In two dimensions, i.e. for curves in R2, we are able to prove that the boundaries of these ω-minima are parameterizable by means of a bi-lipschitz map in a neighbourhood of almost any point of the boundary, i.e. the set of singular points has zero measure with respect toH1 (Th. 6.14). We also show that this regularity result is optimal. In the particular case of minima of (2), when H ∈ L∞(R2), we use the additional regularizing effect of the volume integral to improve the result by showing that the boundary of a minimizer is the graph of a lipschitz function out of a singular set of zero H1-measure (see Prop. 6.5 and Th. 6.19). When the intrinsic unit ball {x : ϕ(x) ≤ 1} is neither a triangle nor a quadrilateral, these regularity results can be improved by showing that the singular set is actually empty (see Th. 6.18). We point out that a regularity result analogous to Proposition 6.5 has been obtained by Morgan et al. [24], in the case of clusters with prescribed volume which minimize (1). The techniques used are similar: comparing a minimizing curve with the segment having the same extremal points. On the other hand, the idea of the proof of Theorem 6.19 is to compute suitable small variations of the functional (2).
The case of dimension greater than two is still open and deserves further investigation. Surprisingly, the usual techniques (i.e. getting the regularity of the boundary from the decay of a suitable notion of excess) seem to fail in this situation (see the second example in Sect. 7).
The plan of the paper is the following:
In Section 2 we fix the intrinsic notation that we will use throughout the paper. In particular, we interpret the functional P (·) in (1) as a kind of perimeter in Rn equipped with a suitable distance related to ϕ. In the case of an even function ϕ, a definition of perimeter equivalent to ours has been given and developed in [4].
In Section 3 we give the definition of ω-minimal set, which extends the notion of minimizer for (2). Moreover, we prove that compactness and density estimates hold for this class of sets (see Prop. 3.2, Prop. 3.3 and Prop. 3.5).
In Section 4 we introduce the notion of excess, that is a quantity which measures the “distance” of the set from being flat in a given ball. In Proposition 4.6 we prove that the boundary of an ω-minimal set coincides with the graph of a lipschitz function up to a set whose (n− 1)-Hausdorff measure is controlled by the excess. In Section 5 we prove that, under the assumption of uniform convexity of the unit ball of the dual space, a polynomial decay of the excess implies theC1,α-regularity of the boundary of ω-minimal sets.
In Section 6 we consider the case when the ambient space is two-dimensional. In particular we show a decay result for the excess (Lem. 6.2) which implies the C1,α-regularity for ω-minimizers under the convexity assump-tion of Secassump-tion 5. We also prove (Th. 6.14) the (local) bi-lipschitz regularity, in the sense of parameterizaassump-tions, for ω-minimizers in two dimensions out of a singular set of zeroH1-measure. In Section 6.4 we show that, when the unit ball of the ambient space is nor a triangle neither a quadrilateral, the set of singular points of the boundary of an ω-minimizer is empty. In Section 6.5 we prove the lipschitz regularity, in the sense of graphs, for the minima of (2), under very general assumption on the shape of the unit ball and on the function H (Th. 6.19).
In Section 7 we provide an example of ω-minimizer in two dimensions which is not (locally) the graph of a lipschitz function, and so we show that Theorem 6.14 is optimal. Moreover, we give an example of global minima for (1) in R3 for which no excess decay property holds. This suggests that, in dimension greater than two, different techniques are needed in order to get a regularity result (if any).
2. Preliminary notions
2.1. Geometric structure
Let X be a real n-dimensional vector space. We say that a function|| · ||: X → R is a general norm if it satisfies
||x + y|| ≤ ||x|| + ||y||, ∀x, y ∈ X, ||λx|| = λ||x||, ∀x ∈ X, ∀λ ≥ 0, ||x|| = 0 ⇐⇒ x = 0, ∀x ∈ X.
We say that|| · || is even if also ||x|| = ||−x|| holds for every x ∈ X (in this case || · || is a norm in the usual sense). The space X endowed with the general norm|| · || will be called a general Minkowski space.
Let X∗be the dual space of X, that is the space of linear functionals on X. If v∈ X and ξ ∈ X∗, we denote by hξ, vi the evaluation of the functional ξ on v. If X is a general Minkowski space the dual space X∗ has the natural general Minkowski structure induced by the general norm || · ||: X∗→ R defined by
||ξ|| := sup
x∈X\{0}
hξ, xi ||x||
(notice that we will use the same symbol|| · || for the general norm of both X and X∗, but the elements of X∗ will always be denoted by Greek letters). For x∈ X, and ρ > 0 we let Bρ(x) :={y ∈ X: ||y − x|| < ρ} be the ball centered in x with radius ρ, we also let Bρ:= Bρ(0). The balls of X∗will be denoted by Bρ∗(ξ) and Bρ∗. In the literature B1 is sometimes called Wulff shape and B∗1 is called Frank diagram.
Let v∈ X \ {0}. We define the orthogonal space v⊥ ⊂ X∗ and the dual face v∗⊂ X∗by
v⊥ := {ξ ∈ X∗:hξ, vi = 0};
v∗ := {ξ ∈ X∗:||ξ|| = ||v||, hξ, vi = ||ξ|| · ||v||} ·
Notice that v∗and v⊥are closed convex, non empty sets. Moreover v∗/||v|| is the sub-differential of the convex function || · ||: X → R at the point v. In the sequel, we will always identify X∗∗ with X (this is done by the canonical isomorphism, which preserves the general Minkowski structures) so that if ξ ∈ X∗\ {0}, the sets ξ⊥ and ξ∗ will be considered as subsets of X. One can also think of ξ∗ as the set of vectors perpendicular to ξ⊥, indeed for all v∈ ξ∗, w∈ ξ⊥ we have||v − w|| ≥ hξ, v − wi/||ξ|| = ||v||.
Given ξ ∈ X∗ \ {0} and v ∈ ξ∗, it is natural to consider the decomposition X = ξ⊥⊕ Rv. So given
f : ξ⊥ → Rv we call graph of f along v the set Γf :={z + tv: z ∈ ξ⊥, tv = f (z)} and subgraph of f along v the set Γ−f :={z + tv: z ∈ ξ⊥, t≤ s, where sv = f(z)}.
Given a function f : ξ⊥ → Rv (ξ ∈ X∗, v∈ ξ∗) we say that f is L-lipschitz if
||f(x) − f(y)|| ≤ L||x − y|| for all x, y ∈ ξ⊥.
We say that the general Minkowski space (X,|| · ||) is euclidean whenever there exists an inner product (·, ·) on
X such that ||x|| =p(x, x) for all x∈ X. In this case v∗ and ξ∗ consist of a single element and v7→ v∗ turns out to be an isomorphism between X and X∗ such thathv∗, wi = (v, w).
2.2. Metric structure and Hausdorff measures
We will consider on (X,|| · ||) the topology induced by the pre-base {Bρ(x): x∈ X, ρ > 0}. It is not difficult to see that this topology does not depend on || · ||, so that it is the usual euclidean topology of X (i.e. the topology induced by the usual topology of Rn through any linear isomorphism between X andRn).
In order to give a structure of metric space to X, it is useful to introduce the symmetrized norm
||x||s:= max{||x||, ||−x||}, which induces on X the distance ds(x, y) =||x − y||s. Note that
diam(E) := sup
x,y∈Eds(x, y) = supx,y∈E||x − y||.
By means of the distance ds we can define on X, for k ∈ [0, n], the Hausdorff measures Hk and the spherical Hausdorff measuresSk (see [16]). Notice that these measures do depend on the general norm|| · || of the space. Anyway, given two different general norms on X, we can find a constant C > 0 such that C−1Hk ≤ ˜Hk ≤ CHk, whereHk and ˜Hk are the two induced Hausdorff measures. This implies thatHk-negligible sets do not depend on|| · ||.
Since we have defined lipschitz functions and Hausdorff measures, we can introduce the notion of k-rectifiable
sets of X as the Hk-measurable sets which are contained, up to a Hk-negligible set, in a countable union of graphs of lipschitz functions fromRk to X. Notice that the family of rectifiable sets does not depend on the general norm of the space X since the metrics dsinduced by different general norms are equivalent.
We recall the main properties of rectifiable sets used in the sequel (here ωk is the usual Lebesgue measure of the euclidean ball ofRk).
Proposition 2.1. Let E⊂ X be k-rectifiable with Hk(E) <∞. Then, denoting by Bρs(x) the balls with respect
to the symmetrized norm, we have
lim ρ→0 Hk(Bs ρ(x)∩ E) ωkρk = 1 for H k-a.e. x∈ E
and Hk(E) =Sk(E).
Proof. The density property is proved in [21] in a more general metric setting. The agreement ofHk andSk is a simple consequence of this fact, arguing as in 3.2.26 of [16].
2.3. Perimeter
On the Borel σ-algebra B(X) is defined a unique Haar measure L invariant under translations and such that L(B1) = ωn. We call this measure, Lebesgue measure on X and we simply write R f (x) dx instead of
R
f (x) dL(x) and |A| instead of L(A). If Ek, E∈ B(X) we write Ek → E as k → ∞ and say that Ek converge to E locally in measure if |(Ek4E) ∩ Bρ| → 0 as k → ∞ for all ρ > 0. Again notice that, given two different general norms on X, it is possible to find a constant C > 0 such that the estimate C−1L ≤ ˜L ≤ CL holds between the induced Lebesgue measures. So the local convergence in measure defined above does not depend on the general norm on X. If k · k is even we also notice that L = Hn; in factHn and L are both invariant Haar measures on X andHn(B1s) = ωn.
With respect to the Lebesgue measure L we can consider the Lebesgue spaces Lp(X) := Lp(X,L) (again, the topology of these spaces does not depend on the general norm on X). If V is a topological vector space and Ω⊆ X is an open set, we can define, as usual, the space of differentiable functions Ck(Ω; V ) andCck(Ω; V ) for
k = 0, . . . , +∞ (if V is not specified we assume V = R). Note that if f ∈ Ck+1(Ω) the differential Df belongs to Ck(Ω; X∗), in fact Df (x) induces the element of X∗ defined by
hDf(x), vi :=∂f
∂v(x) ∀v ∈ X.
We also consider, as usual, the space of distributions D(Ω; V ) as L(Cc∞(Ω), V ) where L(W, V ) is the space of all continuous linear applications L: W → V . If Ω ⊂ X and T ∈ D(Ω; R) we define DT ∈ D(Ω; X∗) by
hhDT, ϕi, vi := −hT, ∂ϕ/∂vi (where ϕ ∈ C∞
If E ⊂ X we denote by χE: X → {0, 1} the characteristic function of E. When E ∈ B(X) and the distributional derivative DχE is a Radon measure on X (a vector measure with values in X∗) we say that E is a set of locally finite perimeter and we define the perimeter of E in any bounded Borel set B as P (E, B) :=
||−DχE||(B). Here ||µ|| represents the total variation of µ, defined by
||µ|| = M−sup
v∈X\{0}
hµ, vi ||v|| ,
where the supremum is taken in the family of all positive measures, with the partial ordering given by
µ≤ ν ⇐⇒ µ(B) ≤ ν(B) ∀B ∈ B(X).
For notational reasons, in the following we write ¯DχE in place of−DχE.
Proposition 2.2. Let A⊂ X be an open set and let E ⊂ X be a set with locally finite perimeter. Then P (E, A) = sup Z Ediv Φ(x) dx: Φ∈ C 1 c(A; X), ||Φ||L∞ ≤ 1 · (3)
Moreover E7→ P (E, A) is lower semicontinuous with respect to local convergence in measure. Proof. Consider the linear functional LE:C0(A; X)→ R induced by the measure ¯DχE A:
LE(Φ) := Z
AΦ(x) d ¯DχE(x).
By Riesz theorem, LE is continuous and||LE|| = || ¯DχE||(A) = P (E, A). Since Cc1(A; X) is dense in C0(A; X) and LE is continuous, we get
||LE|| = sup{LE(Φ): Φ∈ C0(A; X),||Φ||L∞ ≤ 1}
= sup{LE(Φ): Φ∈ Cc1(A; X),||Φ||L∞ ≤ 1}·
Since for Φ∈ C1c(A; X) we have LE(Φ) =REdiv Φ(x) dx, (3) is proved.
In (3) we have represented the perimeter as the supremum of a family (indexed by Φ) of functionals continuous with respect to local convergence in measure; as a consequence the perimeter is lower semicontinuous.
Let E be a set with locally finite perimeter, and let
νE(x) := lim ρ→0+
¯
DχE(Bρ(x))
|| ¯DχE||(Bρ(x))
(if the above limit does not exist at x, we set by convention νE(x) = 0). Then, by Besicovitch differentiation theorem [25] we know that ¯DχE = νE|| ¯DχE|| and νE ∈ L1(X,|| ¯DχE||; X∗) with||νE(x)|| = 1 for || ¯DχE||-a.e. x∈ X.
Then, following [14] we define the reduced boundary to be the set ∂∗E of all x ∈ spt || ¯DχE|| such that ||νE(x)|| = 1. When x ∈ ∂∗E we call νE(x) the exterior unit normal vector to E at x. We also introduce the
density boundary ∂mE := X\ (E0∪ E1) where Eα is defined by
Eα:= x∈ X: lim ρ→0 |E ∩ Bρ(x)| ωnρn = α ∀α ∈ [0, 1].
Notice that ∂mE, i.e. the set of points where the density of E is neither 0 nor 1, does not depend on the
general norm of the space. For ξ ∈ X∗\ {0} define Hξ :={x ∈ X: hξ, xi ≤ 0}, so that νHξ(x) = ξ/||ξ|| for all
On the other hand ∂∗E depends on the general norm of the space. For example consider X =R2,k(x, y)k :=
2
√πmax{|x|, |y|} and E := {(x, y): x ≤ 0, y ≤ 0}. Then one can find that ¯DχE(Bρ) = ρ√2π(1, 1) ∈ X∗ and
k ¯DχEk(Bρ) = ρπ2 (in fact for (ξ, η) ∈ X∗ we have k(ξ, η)k = √2π(|x| + |y|)). So that νE(0) = √1
π(1, 1) and
kνE(0)k = 1 which means 0 ∈ ∂∗E. If instead we consider the usual euclidean norm onR2 then we would find
νE(0) = (1, 1) but|νE(0)| =√2 so that 06∈ ∂∗E.
In the euclidean case, the following fundamental results were proved by De Giorgi and Federer (see [14, 15]).
Theorem 2.3 (De Giorgi–Federer). Let X =Rn endowed with the euclidean norm and let E⊂ X with locally finite perimeter. Then the reduced boundary ∂∗E is (n− 1)-rectifiable, is contained in E1/2 and
P (E, B) =Hn−1(∂∗E∩ B) ∀B ∈ B(X). Moreover Hn−1(∂mE\ ∂∗E) = 0.
We will extend this result to general Minkowski spaces. To this aim, let us consider a linear isomorphism
λ: Y → X, where X and Y are general Minkowski spaces. We recall that the adjoint map λ∗: X∗ → Y∗ is defined by hλ∗ξ, vi = hξ, λvi for all ξ ∈ X∗ and v ∈ Y . If T is a distribution on Y (with values in a vector space V ) we recall that λ#T is the distribution on X (still with values in V ) defined byhλ#T, Φi := hT, Φ ◦ λi
for all Φ∈ Cc∞(X,R). We will say that λ preserves the Lebesgue measure if |λ(B)| = |B| for one (and then for all, provided|B| > 0) Borel set B ⊂ Y . This is equivalent to require that λ#LY =LX, whereLY and LX are respectively the Lebesgue measures on Y and X.
The following result can be proved arguing as in Lemma 3.5 of [20].
Lemma 2.4. Let Y =Rnendowed with the euclidean norm. Let B⊂ Y be a bounded convex open set containing the origin and let F ⊂ Y be a set with locally finite perimeter. Suppose that there exist ρ0> 0 and C > 0 such that
P (F, y + ρB)≤ C|DχF(y + ρB)| ∀ρ ∈ ]0, ρ0[
for some y ∈ spt ||DχF||. Then y ∈ ∂mF .
Theorem 2.5. Let X, Y be general Minkowski spaces, and let λ: Y → X be a linear isomorphism which pre-serves the Lebesgue measure. Then, for all sets E = λ(F )⊂ X with locally finite perimeter,
P (E, B) =
Z
Bϕ(νE(x)) dλ#|| ¯DχF||(x) ∀B ∈ B(X) (4)
where ϕ: X∗→ R is the convex and positively 1-homogeneous function defined for ξ 6= 0 by ϕ(ξ) = ||ξ||2/||λ∗ξ||. Moreover if Y is euclidean we find that
λ(∂∗F )⊂ ∂∗E⊂ ∂mE = λ(∂mF ). (5)
As a consequence ∂∗E is (n− 1)-rectifiable and Hn−1(∂mE\ ∂∗E) = 0. Proof. First of all notice that if f ∈ L1(Y,R) we have
hλ#Df, wi = hD(f ◦ λ−1), λwi = hλ∗D(f◦ λ−1), wi (6)
for all vectors w∈ Y . In fact, given Φ ∈ Cc∞(X,R) we have λ#∂f ∂w, Φ = − Z Y f (y) ∂(Φ◦ λ) ∂w (y) dy =− Z Y f (y) ∂Φ ∂(λw)(λy) dy = − Z X(f◦ λ −1)(x) ∂Φ ∂(λw)(x) dx = ∂(f◦ λ−1) ∂(λw) , Φ ·
So, for f = χF we obtain || ¯DχE|| = M−sup v∈X\{0} h ¯DχE, vi ||v|| =M−supv∈X\{0} h−λ#DχF, λ−1vi ||v|| = M−sup w∈Y \{0} hλ#Dχ¯ F, wi ||λw|| ·
Notice that λ#(ξµ) = (ξ◦ λ−1)(λ#µ) whenever µ is a positive measure; hence for ξ = νF and µ =|| ¯DχF|| we obtain || ¯DχE|| = M−sup w∈Y \{0} h(νF◦ λ−1)(λ#|| ¯DχF||), wi ||λw|| = sup w∈Y \{0} hνF◦ λ−1, wi ||λw|| ! λ#|| ¯DχF||.
But since λ∗ DχE(B)= λ#DχF(B) for any Borel set B, we know that νF◦ λ−1 and λ∗νE have|| ¯DχE||-a.e. the same direction, so
νF◦ λ−1 = λ ∗ν E ||λ∗νE|| || ¯DχE||-a.e. in X. (7) As a consequence || ¯DχE|| = sup w∈Y \{0} hνE, λwi ||λw|| ! λ#|| ¯DχF|| ||λ∗νE|| = ||νE|| ||λ∗νE||λ#|| ¯DχF|| = 1 ||λ∗νE||λ#|| ¯DχF||. That is || ¯DχE|| = (ϕ ◦ νE)(λ#|| ¯DχF||) and (4) is proved.
Now suppose that|| · ||Y =| · |Y is euclidean, y = λ−1(x)∈ ∂∗F , Eρ:= (E− x)/ρ, Fρ:= (F− y)/ρ and H :=
λ(HνF(y)). By the proof of De Giorgi rectifiability theorem (see [20]) we know that P (Fρ, B)→ P (λ−1H, B) for
all balls B andHn−1(∂mF\ ∂∗F ) = 0; we can thus apply Reshetnyak continuity theorem (see for instance [22])
and use (4) to obtain that P (Eρ, B)→ P (H, B) as ρ → 0+. So ¯ DχE(Bρ(x)) || ¯DχE||(Bρ(x))= ¯ DχEρ(B1) || ¯DχEρ||(B1) → Dχ¯ H(B1) || ¯DχH||(B1) = λνF(y) ||λνF(y)|| hence x∈ ∂∗E.
Now we prove the inclusion ∂∗E⊂ ∂mE. Let x = λy∈ ∂∗E and choose C > 0 such that C−1||ξ|| ≤ |λ∗ξ| ≤ C||ξ|| ∀ξ ∈ X∗.
We can find ρ0> 0 such that
P (E, Bρ(x))≤ 2|| ¯DχE(Bρ(x))|| ∀ρ ∈]0, ρ0[ and by (6) and (4) this implies
with B = λ−1(B1). By Lemma 2.4 we obtain that y ∈ ∂mF , hence x ∈ ∂mE (notice that the equality λ(∂mE) = ∂mF is trivial). This concludes the proof of (5).
Finally, since (by Th. 2.3) ∂∗F is (n− 1)-rectifiable and Hn−1(∂mF \ ∂∗F ) = 0 we obtain that ∂∗E is
(n− 1)-rectifiable and Hn−1(∂mE\ ∂∗E) = 0 as well.
By Theorem 2.5 and (7) we infer another representation of the perimeter in terms of the euclidean Hausdorff measure.
Corollary 2.6. Let λ:Rn → X be a linear isomorphism preserving the Lebesgue measure. Then, given B ∈ B(X) and a set E = λ(F ) ⊂ X with locally finite perimeter,
P (E, B) =
Z
λ−1(B)∩∂∗F||(λ
−1)∗ν
F(y)|| dHn−1(y).
In the following theorem we provide one more representation of the perimeter, in terms of the intrinsic Hausdorff measure (see also [9]).
Theorem 2.7. Let E ⊂ X be a set with locally finite perimeter. Then P (E, B) =
Z
∂∗E∩Bψ(νE(x))dH
n−1(x)
where ψ: X∗→ R is the convex and positively 1-homogeneous function defined for ξ 6= 0 by ψ(ξ) :=||ξ|| P (Hξ, B1)
Hn−1(ξ⊥∩ B1)·
Proof. Notice that, by translation invariance, any bounded open set instead of B1 could be considered in the definition of ψ. Consider a linear isomorphism λ:Rn → X which preserves the Lebesgue measure and
F = λ−1(E). Let x∈ λ(∂∗F )⊂ ∂∗E, Bρs(x) :={y ∈ X: ||y − x||s< ρ} and Eρ := (E− x)ρ. We have lim ρ→0+ P (E, Bsρ(x)) ωn−1ρn−1 = limρ→0 P (Eρ, B1s) ωn−1 = P (HνE(x), B1s) ωn−1 = ψ(νE(x))
where the convergence of the limit is guaranteed by the Reshetnyak continuity theorem, as in the proof of Theorem 2.5. So, by the theory of spherical differentiation of measures (see [16], 2.10.19) we proved that
|| ¯DχE|| λ(∂∗F ) = (ψ◦νE)Sn−1 λ(∂∗F ). By since ∂∗E\λ(∂∗F ) isHn−1-negligible we have|| ¯DχE|| λ(∂∗F ) = || ¯DχE|| and the rectifiability of ∂∗E together with Proposition 2.1 give
Sn−1 λ(∂∗F ) =Sn−1 ∂∗E =Hn−1 ∂∗E.
Lemma 2.8. Let u(x) :=||x||. Then ||Du|| = L.
Proof. Since u is a lipschitz function, u is differentiable for a.e. x∈ X. Moreover, using the properties of the
general norm, it is not difficult to show that
∂u
∂v(x)≤ ||v|| ∀v ∈ X,
∂u
∂x(x) =||x||
for any differentiability point x∈ X \ {0}. It follows that ||∇u(x)|| = 1 wherever the differential ∇u(x) exists and x6= 0. So Du = ∇uL and ||Du|| = ||∇u||L = L.
Theorem 2.9. There holds P (B1, X) = nωn.
Proof. We approximate χB1 by continuous functions. Let ϕε: [0,∞[→ R be the continuous function defined by
ϕ(t) := 1 if t≤ 1 1 +1− t ε if 1 < t≤ 1 + ε 0 if t > 1 + ε
and define uε(x) := ϕε(||x||). Since u 7→ || ¯Du||(X) is lower semicontinuous we get
P (B1, X)≤ lim inf
ε→0+ || ¯Duε||(X) = lim infε→0+ || ¯Duε||(B1+ε\ B1).
As uε(x) = (1 + 1/ε− ||x||/ε) in B1+ε\ B1, from Lemma 2.8 we get
P (B1, X)≤ lim inf ε→0+ L(B1+ε\ B1) ε = d dρL(Bρ)|ρ=1 = nωn.
So we proved one inequality. Let now u(x) :=||x||. If ||x|| ≤ 1 we have
u(x)− 1 = −
Z 1
0 χ{y: u(y)<t}(x) dt
whence, by linearity, we have
Du =−
Z 1
0 Dχ{y: u(y)<t}dt in B1.
Given any v∈ B1 and Φ∈ Cc∞(B1) we get ∂u ∂v, Φ =− Z 1 0 ∂ ∂vχ{y: u(y)<t}, Φ dt≤ Z 1 0 || ¯Dχ{y: u(y)<t}|| |Φ| dt.
Since this is true for all Φ and v, we obtain
||Du|| ≤
Z 1
0 || ¯Dχ{y: u(y)<t}|| dt in B1.
Notice that || ¯Dχ{y: u(y)<t}||(B1) = P (Bt, B1) so that
ωn=||Du||(B1)≤ Z 1 0 t n−1P (B 1, X) dt = P (B1, X) n ,
and this concludes the proof.
We notice that, in general, P (B1, X)6= P (−B1, X).
Remark 2.10. Arguing as in Theorem 2.9 it is possible to show that P (E∩ Bρ(x), ∂Bρ(x))≤ d
dρ|E ∩ Bρ(x)|, P (E\ Bρ(x), ∂Bρ(x))≤ d
dρ|E ∩ Bρ(x)| whenever E has finite perimeter in X and t7→ |E ∩ Bt(x)| is differentiable at t = ρ.
2.4. Isoperimetric inequalities
Using Theorem 2.7, the following fundamental results can be recovered from [28] (see also [18, 19], where the symmetry assumption is dropped).
Theorem 2.11 (global isoperimetric inequality). Let E ⊂ X be such that |E| < ∞ and P (E, X) = P (B1, X). Then |E| ≤ |B1| and equality holds if and only if E = B1+ h for some h∈ Rn.
From Theorem 2.11 and Theorem 2.9 it easily follows that for a general set E with|E| < +∞ we have
P (E, X)≥ nωn |E| ωn n−1 n ·
Theorem 2.12 (local isoperimetric inequality). There exists a constant β > 0, depending only on (X,|| · ||), such that P (E, BR) Rn−1 ≥ β |BR∩ E| Rn · |BR\ E| Rn ∀R > 0, E ∈ B(X).
2.5. Non intrinsic notation
Let X be a general Minkowski space and let λ: X → Rn be a linear isomorphism preserving the Lebesgue measure. Let also (·, ·) be the usual euclidean inner product of Rn. Then the adjoint map λ∗:Rn → X∗ verifies hλ∗ξ, λ−1vi = (ξ, v). So we may define the functions ϕ: Rn → R, ϕ(v) := ||λ−1v||, ϕo:Rn → R,
ϕo(ξ) =||λ∗ξ||, T : Rn → P(Rn), T (v) := λ∗−1 (λ−1v)∗, To:Rn → P(Rn), To(ξ) := λ (λ∗ξ)∗, Wϕ:= λ(B1),
Fϕ:= (λ∗)−1(B1∗), Pϕ(E, B) := P (λ(E), λ(B)). This provides an equivalent definition of the space (X,|| · ||) as Rn endowed with the general norm ϕ, which is the usual approach followed in the literature [9, 26].
By Corollary 2.6, we have P (E, B) = Z λ(∂E∩B)ϕ o ν λ(E)(x)dHn−1(x) where νλ(E)(x) is the usual euclidean normal vector to λ(E) in x.
3. ω–minimal sets
In order to have boundary, closure and interior operators invariant under changes in negligible sets, we define
∂E := {x ∈ X: ∀ρ > 0 |E ∩ Bρ(x)| 6∈ {0, |Bρ(x)|}} ,
E := {x ∈ X: ∀ρ > 0 |E ∩ Bρ(x)| 6= 0} , ˚
E := {x ∈ X: ∃ρ > 0 |E ∩ Bρ(x)| = |Bρ(x)|} ·
Notice that if |E4F | = 0 then ∂E = ∂F , E = F and ˚E = ˚F . Moreover ∂E, E are closed sets, ˚E is open and
the usual relations ∂E = E\ ˚E and ˚E = X\ X \ E hold. Notice that in general |E \ ˚E|, |E \ E| > 0. If Ω ⊂ X
we write E b Ω when E ⊂ Ω and E is compact.
Let ω: (0,∞) → (0, ∞] be a nondecreasing function with limρ→0+ω(ρ) = 0.
Definition 3.1. Let Ω⊂ X be an open set. We say that E ⊂ X is an ω-minimal set in Ω if for all Bρ(x)b Ω with x∈ ∂E and all F ⊂ X with E4F b Bρ(x) we have
P (E, Bρ(x))≤ (1 + ω(ρ))P (F, Bρ(x)).
Proposition 3.2 (density upper bound). There exists a constant Θ > 0 (depending only on n, X, ω) such that if E∈ Mω(Ω) and Bρ(x)b Ω with ω(ρ) ≤ 1 then
P (E, Bρ(x))≤ Θρn−1.
Proof. Suppose for simplicity x = 0. Take η < ρ and let Fη= E∪ Bη. Then we have
P (Fη, Bρ) = P (E, Bρ\ Bη) + P (Fη, ∂Bη)≤ P (E, Bρ\ Bη) + nωnηn−1
since P (Fη, ∂Bη)≤ P (Bη, X) = nωnηn−1. Then, since Fη4E b Bρ we get
P (E, Bρ)≤ (1 + ω(ρ)) P (E, Bρ\ Bη) + nωnηn−1
and letting η→ ρ− we conclude
P (E, Bρ)≤ nωn(1 + ω(ρ))ρn−1.
Using Theorem 2.7 and reasoning as in [3, 3.2], one can get the following compactness result (see also [5]).
Proposition 3.3 (compactness of ω-minimal sets). Let (Ek) be a sequence of ωk-minimal sets in Ωk⊂ X and suppose that ωk → ω pointwise and Ωk ↑ X. Then there exists a subsequence (Ekj) converging in L1loc(X) to
E ∈ Mω. Moreover, if Ek converge in L1loc(X) to E then
DχEk* Dχ∗ E and || ¯DχEk||*∗ || ¯DχE||.
Proposition 3.4 (volume bounds). Let E ∈ Mω(Ω). Then there exists a constant γ ∈]0, 1[, depending only
on X and ω, such that
γ≤ |E ∩ Bρ(x)|
ωnρn ≤ 1 − γ whenever, Bρ(x)b Ω, ω(ρ+) < 1 and x∈ ∂E.
Proof. Suppose for simplicity x = 0 and consider the nondecreasing function g(ρ) :=|E ∩ Bρ|. Theorem 2.11
gives
1
C1g(ρ) n−1
n ≤ P (E ∩ Bρ, X) = P (E, Bρ) + P (E∩ Bρ, ∂Bρ)
for some dimensional constant C1> 0. By Remark 2.10 we find that P (E∩ Bρ, ∂Bρ), P (E\ Bρ, ∂Bρ)≤ g0(ρ) for all ρ > 0 for which g0(ρ) exists. Comparing E with E∩ Bρ we have
P (E, Bη)≤ (1 + ω(η))P (E \ Bρ, Bη) ∀η > ρ and letting η→ ρ+, if ω(ρ+) < 1 we get
P (E, Bρ)≤ (1 + ω(ρ+))P (E\ Bρ, Bρ) = (1 + ω(ρ+))P (E\ Bρ, ∂Bρ)≤ 2g0(ρ).
To prove the second inequality, consider X\ E instead of E and repeat the proof; even if, in general, we cannot assume that X\ E is ω-minimal, the inequality
P (X\ E, Bρ)≤ CP (E, Bρ) is sufficient to complete the proof.
Notice that the previous estimate ensures that if xk ∈ ∂Ek, Ek→ E in L1loc(X), xk→ x and Ek ∈ Mωthen
x∈ ∂E. This will be often used in the following:
Proposition 3.5 (density lower bound). There exist θ > 0 and ρω> 0 such that, if E∈ Mω(Ω) and Bρ(x)b Ω with ρ < ρω and x∈ ∂E, then
P (E, Bρ(x))≥ θρn−1. (8)
Proof. In the case E∈ M0(Ω) we will reason as in [3, 3.4], assuming for simplicity n > 2 (in dimension n = 2 the proof is quite simpler). Let γ := ((n− 1)n−1ωn−1)n−21 and consider a bijection λ: X → Rn as in Section 2.5.
As a consequence of the isoperimetric inequalities for subsets ofRn of arbitrary codimension (see [1]), it follows that for a.e. η > 0 there exists a set λ(Eη)⊂ Rn with locally finite perimeter, such that λ(E)4λ(Eη)⊆ Bη(λx) and
P (λ(Eη), Bη(λx))≤ γ m0(η)
n−1
n−2, where m(η) := P (λ(E), B
η(λx)). Hence, setting c := max{sup|e|=1ϕo(e), (inf|e|=1ϕo(e))−1}, by the minimality of E we get
c−1m(η)≤ P (E, λ−1(Bη(λx)))≤ P (Eη, λ−1(Bη(λx)))
≤ cP (λ(Eη), Bη(x))≤ cγ(m0(η))
n−1 n−2,
for a.e. η > 0. This implies
m(η) ηn−1 ≥ 1 (n− 1)n−1 1 2γc2 (n−1)2 n−2 ,
and letting η = c−1ρ we get λ(Bρ(x))⊇ Bη(λx) so that
P (E, Bρ(x)) ρn−1 ≥ P (λ(E), Bη(λx)) cnηn−1 ≥ 1 cn(n− 1)n−1 1 2γc2 (n−1)2 n−2 =: 2θ.
In the general case E ∈ Mω(Ω), we argue as in [8]. Suppose, by contradiction, that there exist sets Ek∈ Mω(Ω), points xk∈ ∂Ek and radii ρk→ 0 such that
ρ1−nk P (Ek, Bρk(xk)) < θ.
By the rescaling properties of quasi-minimizers the sets Fk:= (Ek−xk)/ρkbelong toMωk((Ω−xk)/ρk), where
ωk(t) := ω(ρkt). We note that ωk → 0 so that by Proposition 3.3, up to a subsequence, we may suppose that Fk → F with F ∈ M0. From the lower semi-continuity of perimeter we obtain
θ≥ lim inf
k→∞ ρ
1−n
k P (Ek, Bρk(xk)) = lim inf
k→∞ P (Fk, B1)≥ P (F, B1).
But since by Proposition 3.4 0∈ ∂F and F ∈ M0 we have already proved that P (F, B1)≥ 2θ, so that we have a contradiction.
Corollary 3.6. For any E∈ Mω(Ω), there holds
Hn−1 (∂E∩ Ω) \ ∂∗E) = 0.
Proof. By (8) the measure k ¯DχEk has strictly positive (spherical) (n − 1)-dimensional upper density greater than θ/ωn−1at any x∈ ∂E, hence
k ¯DχEk(B) ≥ θ ωn−1H
n−1(B) ∀B ∈ B(X), B ⊂ ∂E.
Choosing B = ∂E\∂∗E and taking into account thatk ¯DχEk is concentrated on ∂∗E the conclusion follows.
By Corollary 3.6, when we deal with ω-minimal sets, we can equivalently integrate with respect toHn−1(or
|| ¯DχE||) either on ∂E or on ∂∗E.
Remark 3.7 (additive quasi minimizers). Let us define an additive ω-minimal set in Ω as a set E ∈ B(X) such
that
P (E, Bρ(x))≤ P (F, Bρ(x)) + ρn−1ω(ρ)
for any ball Bρ(x)b Ω and any F ⊂ X with E4F b Bρ(x). If E satisfies also (8) for some θ > 0, it is easy to check that E is also a ω/(θ− ω)-minimizer in the sense of Definition 3.1.
An example of sets which satisfy the additive ω-minimal condition and the density lower bound (8) is given by sets with prescribed mean curvature in Ln, that is, the minima of the functional
E7→ P (E, Ω) +
Z
E∩ΩH(x) dx with H∈ Ln(Ω) (see [8]).
Remark 3.8. Let λ : X→ Rnbe as in Corollary 2.6, let E∈ Mω(Ω) with ∂E∩Ω 6= ∅ and set E0:= λ(E)⊂ Rn, Ω0 := λ(Ω). Then, for any δ > 0 there exists a constant k > 0, depending only on ω and δ, such that E0 is a (Ω0, k, δ)-minimizer in the sense of [13]. More precisely, E0 satisfies the following properties:
1. ∂E0∩ Ω06= ∅;
2. P (E0, B) < +∞ for each B b Ω0;
3. P (E0, Ω0)≤ kP (ϕ(E0), Ω0) whenever ϕ∈ Lip(Rn;Rn), diam(W ∪ ϕ(W )) < δ and W ∪ ϕ(W ) b Ω0b Rn, where W :={x ∈ Rn : ϕ(x)6= x}.
By the previous remark and [13] (Th. 2.11) it follows that ∂E0 fulfils some mild regularity properties. Precisely, for any x∈ ∂E0 and R∈]0, δ[ such that B3R(x)⊂ Ω0, there exists a constant C > 0, depending only on ω and
δ, such that for any y∈ ∂E0∩ BR(x) and any ball Br(y)⊂ BR(x) there exists a C-lipschitz graph Γ such that
Hn−1(∂E0∩ Γ ∩ B
r(y))≤ 1
Cr
n−1.
Being the statement bi-lipschitz invariant, the same holds for ∂E, by composing with λ−1.
Lemma 3.9 (cut and paste). Let Ω⊂ X be an open set and let E, H ⊂ X be sets with locally finite perimeter. Suppose moreover that E ∩ ∂Ω = H ∩ ∂Ω, and Hn−1(∂E∩ ∂Ω) = Hn−1(∂H∩ ∂Ω) = 0. Then, letting F := (E\ Ω) ∪ (H ∩ Ω) we have
P (F, B) = P (E, B\ Ω) + P (H, B ∩ Ω) for all Borel subsets B⊂ X.
In particular, if E∈ Mω and Ωb Bρ(x0) with x0∈ ∂E, we have
P (E, Ω)≤ P (H, Ω) + ω(ρ+)P (F, Bρ(x0)) = 1 + ω(ρ+)P (H, Ω) + ω(ρ+)P (E, Bρ(x0)\ Ω).
Proof. First we prove that ∂F ∩ ∂Ω ⊂ ∂E ∪ ∂H. Let x ∈ ∂Ω \ (∂E ∪ ∂H) and suppose x ∈ ˚E (otherwise
we may consider the complementary sets of E and H). By hypothesis x ∈ ˚H so we get, for some ρ > 0, |Bρ(x)∩E| = |Bρ(x)∩H| = ωnρn. Then obviously|Bρ(x)∩F | = ωnρn, which means x6∈ ∂F . So we have proved that ∂F∩ ∂Ω ⊂ ∂E ∪ ∂H and in particular P (F, ∂Ω) = 0 since Hn−1(∂F∩ ∂Ω) ≤ Hn−1((∂E∪ ∂H) ∩ ∂Ω) = 0.
So, by the locality of perimeter, for all Borel sets B⊂ X we have
P (F, B) = P (F, B∩ Ω) + P (F, B \ Ω) + P (F, B ∩ ∂Ω)
= P (H, B∩ Ω) + P (E, B \ Ω).
Now let us prove the second statement. Suppose for simplicity x0 = 0. Choose a radius η > ρ such that
P (E, ∂Bη) = 0. Since E4F ⊂ Ω b Bη from the minimality of E we get P (E, Bη)≤ (1 + ω(η))P (F, Bη). But since we have P (E, Bη) = P (E, Ω) + P (E, Bη\ Ω) and P (F, Bη) = P (H, Ω) + P (E, Bη \ Ω), the conclusion follows letting η→ ρ+.
Proposition 3.10 (Gauss–Green). Let B be an open set and E, F sets of X with locally finite perimeter in X. If E4F b B then DχE(B) = DχF(B). The same result holds if we know that E∩ ∂B = F ∩ ∂B and
Hn−1(∂F∩ ∂B) = Hn−1(∂E∩ ∂B) = 0.
Proof. Given ϕk ∈ Cc∞(B) such that ϕk = 1 on E4F and ϕk → χB in L1 we have
|hDχE− DχF, ϕki| = Z EDϕk− Z FDϕk ≤Z E4F|Dϕk| = 0 and the first result follows.
For the second statement, we set F0 := (F ∩ B) ∪ (E \ B). We notice that F04E ⊂ B. For any ρ > 0 consider the ρ-neighbourhood of B: Bρ := {x ∈ X: infy∈B||x − y|| ≤ ρ}. Since F04E b Bρ we know that DχE(Bρ) = Dχ0F(Bρ) and letting ρ → 0+ we obtain DχE(B) = DχF0(B). Since P (E, ∂B) = 0 and F0∩ B = F ∩ B we conclude DχE(B) = DχF0(B) = DχF(B).
4. Lipschitz approximation of sets with small excess
In this section we introduce the notion of excess, that is a quantity which measures the “distance” of the set from being flat in a given ball. In Proposition 4.6 we will prove that the boundary of an ω-minimal set coincides with the graph of a lipschitz function up to a set whose (n− 1)-Hausdorff measure is controlled by the excess.
In order to study the regularity of ω-minimal sets we consider the following quantities (see [5])
EccvE(x, ρ) := ρ1−n || ¯DχE||(Bρ(x))− h ¯DχE(Bρ(x)), vi, EccE(x, ρ) := inf
v∈X, ||v||=1Ecc v
E(x, ρ) = ρ1−n || ¯DχE||(Bρ(x))− || ¯DχE(Bρ(x))||. We also define the singular set Σ(E) as
Σ(E) := (
x∈ ∂E : lim sup
ρ→0+ EccE(x, ρ) > 0
)
Notice that Σ(E)∩ ∂∗E =∅; indeed if x ∈ ∂∗E we have, by definition, lim ρ→0+ ¯ DχE(Bρ(x)) || ¯DχE||(Bρ(x)) = νE(x) where ||νE(x)|| = 1; i.e. lim ρ→0+ || ¯DχE||(Bρ(x))− || ¯DχE(Bρ(x))|| || ¯DχE||(Bρ(x)) = 0.
Since by (5) x∈ ∂mE, the relative isoperimetric inequality implies thatk ¯DχEk(Bρ(x))≥ cρn−1 for a suitable
c > 0 and ρ sufficiently small. We conclude that limρ→0+EccE(x, ρ) = 0. Since ∂∗E and Σ(E) are disjoint it
follows thatk ¯DχEk(Σ(E)) = 0 and Theorem 2.7 gives
Hn−1(Σ(E)) = 0. (9)
These quantities are meant to measure the “flatness” of the set E in the ball Bρ(x). In fact, when B∗1 is strictly convex and EccE(x, ρ) = 0 then E∩ Bρ(x) is an half-plane. If B1∗is not strictly convex this is not always true, but we can state the following result:
Lemma 4.1. Let x0∈ X, v ∈ ∂B1and let T ⊂ X be a hyper-space perpendicular to v. If EccvE(x0, ρ) = 0 then E coincides in Bρ(x0) with the subgraph of a L0-lipschitz function f : T → Rv. The constant L0 depends only on (X,|| · ||).
Proof. Let us choose mollifiers ϕε ∈ C0∞(X) such that spt ϕε ⊂ Bε, ϕε ≥ 0, RBεϕε = 1 and ϕε(x) = 3 (4ωn(ε/2)n)−1 for x∈ Bε/2. We consider the mollified functions uε:= χE∗ ϕε.
Let x∈ X, ε > 0 be such that uε(x) = 1/2 and Bε(x)⊆ Bρ(x0). Then 3 4 |E ∩ Bε/2(x)| |Bε/2| = Z E∩Bε/2(x) ϕε(y− x) dy = Z Eϕε(y− x) dy − Z E∩(Bε(x)\Bε/2(x)) ϕε(y− x) dy ≥ 1 2 − 1−3 4 =1 4
that is |E ∩ Bε/2(x)| ≥ 13|Bε/2(x)|; reasoning the same way with X \ E we also find |Bε/2(x)\ E| ≥ 13|Bε/2(x)|. Hence, by Theorem 2.12 we get P (E, Bε/2(x)) > α(ε/2)n−1, with α > 0 depending only on (X,|| · ||).
Consider now the positive measures µε:= ϕε(· − x)|| ¯DχE||. Letting αε:= µε(Rn), for ε sufficiently small we have αε≥ µε(Bε 2(x)) = 3 4ωn(ε/2)nP (E, Bε/2(x))≥ 3α 2ωnε ≥ α.
Notice that since EccvE(x0, ρ) = 0, we have P (E, Bρ(x0)) =
Z Bρ(x0)
hνE(y), vi d|| ¯DχE||(y),
which means that νE(y)∈ v∗ for|| ¯DχE||-a.e. y ∈ Bρ(x0). Hence, since µε/αεare probability measures and v∗ is a convex set, we obtain
¯ ∇uε(x) = ( ¯DχE∗ ϕε)(x) = αε Z νE(y) dµε αε(y)∈ αεv ∗.
We have just proved that if uε(x) = 1/2 (with Bε(x)⊆ Bρ(x0)) then ¯Duε(x)∈ αεv∗with αε≥ α, which means that ∂uε/∂v(x) < 0 ashξ, vi > 0 for all ξ ∈ v∗. Thus we may apply the implicit function theorem to obtain that
{uε(x) = 1/2} ∩ Bρ−ε(x0) is contained in the graph of a C∞ function fε: T → Rv. Since εk∇uεk is bounded from above in X and ε∂uε/∂v is bounded from below in{uε= 1/2} we obtain that
∂fε ∂z =− ∂uε ∂v −1∂u ε ∂z z∈ T ∩ ∂B1
is bounded, hence fεare L0-lipschitz for some constant L0 independent of ε.
Since uε→ χEin L1, letting Eε:={uε≥ 1/2} we have |Eε4E| → 0. Recalling that ||Dfε|| are equi-bounded, we have
||fε1− fε2||C0 ≤ C||fε1− fε2||L1= C|Eε14Eε2|
for some dimensional constant C, so fε → f uniformly and since fε are all L0-lipschitz also f is L0-lipschitz and E∩ Bρ(x0) is the subgraph of f along v.
Proposition 4.2. The best constant L0 in the previous theorem, valid for any v ∈ X and any hyper-space T perpendicular to v, is given by
L0= sup
||v||=1ξ, ν∈vsup∗ z∈ξ⊥sup, ||z||=1|hz, νi| .
In particular, L0= 0 if B1∗ is strictly convex (since in this case ξ = ν in the previous formula).
Proof. Given any v∈ X with ||v|| = 1 the sets E such that EccvE(x, ρ) = 0 are exactly those for which νE ∈ v∗
|| ¯DχE||-a.e. in Bρ(x). So, given ξ∈ v∗ and T = ξ⊥, if f is the lipschitz function given by the previous lemma then we have Df (z)h = −hh, νE(z + f (z))iv (this can be easily proven noticing that {h + tv: h ∈ ξ⊥, tv ≤ Df (z)h} = {h + tv : hh + tv, νE(z + f (z))i ≤ 0}). So the lipschitz constant of Df(z) is given by
sup
||h||=1||Df(z)h|| = sup||h||=1|hh, νi|. The conclusion follows considering all possible v, ξ and ν.
Proposition 4.3 (properties of the excess).
1. If Bρ(x)⊂ Bη(y), then EccvE(x, ρ) ≤ η ρ n−1 EccvE(y, η) EccE(x, ρ) ≤ η ρ n−1 EccE(y, η).
2. The functions ρ7→ EccE(x, ρ) and ρ7→ EccvE(x, ρ) are left-continuous in ]0, +∞[. 3. Let (Ek) be a sequence of ω-minimal sets converging to E in L1loc(X) and vk→ v. Then
lim k→∞EccEk(x, ρ) = EccE(x, ρ) lim k→∞Ecc vk Ek(x, ρ) = Ecc v E(x, ρ)
for all x∈ X and all ρ > 0 such that P (E, ∂Bρ(x)) = 0. While lim inf k→∞ EccEk(x, ρ) ≥ EccE(x, ρ) lim inf k→∞ Ecc vk Ek(x, ρ) ≥ Ecc v E(x, ρ),
for all x∈ X and all ρ > 0.
4. If EccE(x, ρ) = 0 then E ∈ M0(Bρ(x)).
Proof. 1. Note that
ρn−1EccvE(x, ρ) = Z
Bρ(x)
(1− hνE(y), vi)d|| ¯DχE||(y).
Since the integrand is always nonnegative, we can conclude that ρn−1EccvE(x, ρ)≤ ηn−1EccvE(y, η). Moreover, if EccE(x, ρ) = EccvE(x, ρ) and EccE(y, η) = EccwE(y, η), then
EccE(x, ρ) = EccvE(x, ρ)≤ EccwE(x, ρ)≤ η ρ n−1 EccwE(y, η) = η ρ n−1 EccE(y, η).
2. It follows immediately from the definitions of EccvE and EccE. 3. As DχEk
∗
* DχE and || ¯DχEk||
∗
*|| ¯DχE||, the first statement follows. Given x ∈ X and ρ > 0, choose
ρ0 < ρ such that P (E, ∂Bρ0(x)) = 0. Then, we have
(ρ0)n−1EccE(x, ρ0) = lim h→∞(ρ
0)n−1Ecc
Eh(x, ρ0)≤ lim infh→∞ ρn−1EccEh(x, ρ),
(ρ0)n−1EccvE(x, ρ0) = lim h→∞(ρ
0)n−1Eccv
Eh(x, ρ0)≤ lim infh→∞ ρn−1EccvEh(x, ρ).
One gets the thesis using property 1 and letting ρ0→ ρ.
4. Given any F such that E4F b Bρ(x) by Proposition 3.10 we obtain
P (E, Bρ(x)) =|| ¯DχE(Bρ(x))|| = || ¯DχF(Bρ(x))|| ≤ P (F, Bρ(x)).
Lemma 4.4 (vertical gap). For any L > L0 there exists ε1 = ε1(n,|| · ||, ω, L) > 0 such that the following
property holds. Let E∈ Mω(B3ρ), ω(ρ)≤ 1, v ∈ X ∩ ∂B1 and let T ⊂ X be an hyper-space perpendicular to v. Let x = z + tv, x0= z0+ t0v be two points of Bρ∩ ∂E with z, z0∈ T and t, t0∈ R. If we in addition suppose that
EccvE(w, η)≤ ε1 ∀w ∈ {x, x0} ⊂ Bρ(x0)∩ ∂E, η ∈]0, 2ρ]
we conclude that
||tv − t0v|| ≤ L||z − z0||.
Proof. Without loss of generality we may suppose that ρ = 1. Suppose also, by contradiction, that there
exist ω-minimal sets Ek in B3, vk ∈ ∂B1, hyper-planes Tk and points xk = zk+ tkvk, x0k = z0k+ t0kvk with
xk, x0k∈ B1∩ ∂Ek, zk, zk0 ∈ Tk and tk, t0k∈ R such that Eccvk
Ek(w, η)≤
1
for all η∈ (0, 2], w ∈ {xk, x0k} and
||tkv− t0kv|| > L||zk− zk0||.
Case 1. Suppose that lim infk→∞|tk− t0k| > 0. Up to a subsequence we may suppose that tk → t, t0k→ t06= t, vk → v, Tk → T , xk → x = z + tv, x0k → x0 = x0+ t0v and by Proposition 3.3 Ek → E. Now for k sufficiently
large we have Eccvk Ek(x, 3/2)≤ 4n−1 3n−1Ecc vk Ek(xk, 2)≤ 4n−1 3n−1k·
By Proposition 4.3 we get EccvE(x, 3/2) = 0 and similarly EccvE(x0, 3/2) = 0. So by Lemma 4.1 we know that in
the union B3/2(x)∪ B3/2(x0) the set ∂E coincides with the graph along v of a L0-lipschitz function f : T → Rv. But f (z) = tv and f (x0) = t0v and since we have assumed||tv − t0v|| > L||z − z0|| we get a contradiction when L > L0.
Case 2. Suppose that lim infk→∞|tk− t0k| = 0 and let ηk :=||tkv− t0kv||. Up to a subsequence we may suppose
that ηk → 0. Define Fk := (Ek− xk)/ηk and yk = (xk− x0k)/ηk. We get
||yk|| ≤ ||zk− z 0 k|| + ||tkv− t0kv|| ηk ≤ 1 + 1 L,
so, up to a subsequence, we may suppose yk → y = z ± v/|| ± v|| for some z ∈ T . By the hypothesis
||tkv− t0kv|| > L||zk − zk0|| we get ||z|| < 1/L. Let R := 1 + 1/L. Again, up to a subsequence, we may suppose that Fk → F and that Rηk < 2. The hypothesis Eccvk
Ek(w, Rηk) ≤ 1/k for w ∈ {xk, x 0 k} becomes Eccvk Fk(0, R) ≤ 1/k and Ecc vk Fk(yk, R) ≤ 1/k, so that Ecc vk
Fk(y, R/2) ≤ 2n−1/k, and by Proposition 4.3 we
conclude
EccvF(0, R) = EccvF(0, R/2) = 0.
As in Case 1 we note that the set ∂F must be a graph along v of an L0-lipschitz function f defined on T , but again we note that since f (0) = 0 and f (z) = ±1 the lipschitz constant of f is at least L. This contradicts
L > L0.
Lemma 4.5 (horizontal translations). There exist positive constants c0, ε0depending on (n,|| · ||, ω) such that, given any E∈ Mω(B2(1+L0)ρ(x0)), ω(ρ)≤ 1, and given an hyper-space T and a unit vector v perpendicular to
T , then for all h∈ T ∩ Bρ there holds
EccvE(x0, 2(1 + L0)ρ) < ε0⇒ Hn−1(∂E∩ C(x0, h, ρ))≥ c0ρn−1, where
C(x0, h, ρ) ={x0+ z + tv: z∈ Bρ(h)∩ T, ||tv|| ≤ 2L0ρ} ·
Proof. Suppose for simplicity x0= 0, ρ = 1 and let C(h) := C(0, h, 1). Reasoning by contradiction we suppose that there exist Ek ∈ Mω(B2(1+L0)), hyper-spaces Tk, unit vectors vk perpendicular to Tk and hk ∈ Tk∩ B1 such that Eccvk Ek(0, 2(1 + L0)) < 1 k, and H n−1(∂E∩ C(h k)) < c0.
Up to a subsequence we may also suppose that Ek→ E, vk → v, Tk → T , hk → h, so that EccvE(0, 2(1+L0)) = 0 and Hn−1(∂E∩ C(h)) < c0. So by Lemma 4.1, in B2(1+L0)∂E is the graph over T of an L0-lipschitz function
f : T → Rv. Since C(h) ⊂ B2(1+L0)we get ∂E∩C(h) = Γf∩C(h). Moreover, if π: X → T is the projection on T along v (π(z + tv) := z for all z∈ T , t ∈ R), it is not difficult to show that z ∈ π(C(h)) ⇒ z +f(z) ∈ C(h) which means π(Γf ∩ C(h)) = π(C(h)) = B1(h). Finally we conclude thatHn−1(∂E∩ C(h)) ≥ KHn−1(T ∩ B1(h)), where K is the lipschitz constant of π, so that for c0 sufficiently small we get a contradiction.
In the following proposition, we will show that the boundary of a quasi minimizer E can be locally approxi-mated by the graph Γ of a lipschitz function, estimating the measure of ∂E4Γ with the excess.
Proposition 4.6 (lipschitz approximation). For each L > L0there exists a constant c(L) > 0 such that, given any unit vector v ∈ X ∩ B1 and any hyper-space T ⊂ perpendicular to v, for any E in Mω(Ω), x0∈ ∂E and
for all ρ > 0 sufficiently small (more precisely 9(2L0+ 1)ρ < ρω, ω((2L0+ 1)ρ) < 1 and B9(2L0+1)ρ(x0)⊂ Ω)
the following statements are true.
1. If EccvE(x, η)≤ ε1(L) (where ε1(L) is the constant given in Lemma 4.4) for all x∈ ∂E ∩ Bρ(x0) and for
all η∈ ]0, 2ρ[, then ∂E ∩ Bρ(x0) is contained in the graph along v of a L-lipschitz function f : T → Rv. 2. There exists a L-lipschitz function f : T → Rv such that
Hn−1((∂E4Γ
f)∩ Bρ(x0))≤ c(L)ρn−1EccvE(x0, 9(2L0+ 1)ρ).
Proof. Suppose for simplicity x0= 0.
Step 1. Define
G :=x∈ ∂E ∩ Bρ: EccvE(x, η)≤ ε1(L) ∀η ∈ ]0, 2ρ[ ·
From Lemma 4.4 we know that given any two points x = z +tv and x0 = z0+t0v of G we have|t−t0| ≤ L||z−z0||.
It follows that the projection π of G on T along v is injective and G is the graph of a L-lipschitz function f which can be extended to all T . So the first statement is easily proved, since in that case G = ∂E∩ Bρ.
Step 2. Let now
G0 :=x∈ B3(2L0+1)ρ∩ ∂E: EccvE(x, η)≤ ε ∀η ∈ (0, 6(2L0+ 1)ρ) where ε > 0 is such that ε < ε0 and ε < ε1(L).
Now we will estimate the measure of the set U := ∂E ∩ B3(2L0+1)ρ \ G0. For all x ∈ U there exists
ρx ∈ (0, 6(2L0+ 1)ρ) such that EccvE(x, ρx) > ε, By Besicovitch covering theorem we can find a countable covering {Bρi(xi)} of U (ρi := ρxi), with the property that for all x ∈ U there are at most N balls Bρi(xi) which contain x (N is a constant depending only on (X,k · k)). So we get
Hn−1(U ) ≤ X i Hn−1(∂E∩ B ρi(xi))≤ K X i ρn−1i (10) ≤ K ε X i ρn−1i EccvE(xi, ρi)≤KN ε (9(2L0+ 1)ρ) n−1Eccv E(0, 9(2L0+ 1)ρ).
Step 3. Let f be the L-lipschitz function, defined as in Step 1, such that G0 ⊆ Γf. Also we can suppose
L < 5L0/4. We will now estimate the measure of the set V := (Γf∩ Bρ)\ ∂E. Let π(V ) be the projection of
V on T along v. By (10), it is enough to estimate from above the (n− 1)-dimensional measure of π(V ) with a
multiple of Hn−1(U ). Observe that we can assume G0∩ (−Bρ
3)6= ∅, indeed if not it would follow H
n−1(U )≥
Hn−1(∂E∩(−Bρ
3))≥ c(ρ/3)
n−1, by Proposition 3.5; therefore, by (10) we would have Eccv
E(0, 9(2L0+1)ρ)≥ kε,
for some k > 0, so that choosing c(L) large enough, T itself satisfies the thesis. So we assume G0∩ Bρ 3 6= ∅.
Let z∈ π(V ) and let Cr(z) := (Br(z)∩ T ) ⊕ Rv be the largest cylinder not intersecting G0. We have r < 4ρ/3, moreover there exists z0 ∈ T ∩ ∂Br(z) such that z0 + f (z0) ∈ G0. By Lemma 4.5 with h = z0− z, we get
Hn−1(∂E∩ Cv
r) ≥ c0rn−1, where Crv := C(z + f (z0), z0 − z, r). Since ||z + f(z0)|| ≤ ρ(4 + 8L)/3, it follows that, for L < 5L0/4 we have ∂E∩ Crv ⊂ U ∩ Cr(z), which in turn implies Hn−1(U ∩ Cr(z)) ≥ c0rn−1. As before, we can find a collection of balls{Bri(zi)}, zi∈ T , such that NHn−1(SiBri(zi)∩ T ) ≥ Hn−1(π(V )) and
Hn−1(U∩ C
ri(zi))≥ c0rn−1i . Summing up over i we getHn−1(π(V ))≤ MHn−1(U ), for a suitable constant
We conclude this section recalling the following lemma, that we will be needed in the sequel (see [5], Lem. 4.4.5).
Lemma 4.7. Let E ⊂ X be a set of finite perimeter in Ω, where Ω is an open connected subset of X. If ∂E ∩Ω is non empty and is contained in a lipschitz graph Γ, then ∂E∩ Ω = Γ ∩ Ω.
5. The uniformly convex case
We say that the ball B∗1 is uniformly convex if there exists a constant M > 0 such that for all ξ, η∈ ∂B1∗we have ||ξ − η||2 s≤ M 1−||ξ + η|| 2 ·
For any set E with locally finite perimeter in Ω and any ball Bρ b Ω we define the approximate normal
νE(x, ρ) by
νE(x, ρ) := Dχ¯ E(Bρ(x))
|| ¯DχE(Bρ(x))||·
Proposition 5.1. Assume that B1∗ is uniformly convex and let E ∈ Mω(Ω) be such that EccE(x, ρ) ≤ Cρ2α
for all balls Bρ(x)⊂ Ω, x ∈ ∂E, for some constant C, α > 0. Then ∂∗E = ∂E in Ω and νE(x) : ∂E∩ Ω → X∗
is α-H¨older continuous. Hence, ∂E∩ Ω is an hyper-surface of class C1,α.
Proof. From the lower density estimate and the upper bound on the excess we get || ¯DχE(Bρ(x))|| ≥ 2θρn−1 ∀Bρ(x)⊂ Ω, ρ ∈ ]0, ρ0[ , for a suitable constant ρ0.
Let Bρ(x)⊂ Bη(y)⊂ Ω with x, y ∈ ∂E and η < ρ0; for any w∈ νE(y, η)∗ we obtain
2||νE(x, ρ)− νE(y, η)||2s ≤ M(2 − ||νE(x, ρ) + νE(y, η)||) ≤ M(2 − hνE(x, ρ) + νE(y, η), wi) = M (1− hνE(x, ρ), wi) ≤ M || ¯DχE||(Bρ(x)) || ¯DχE(Bρ(x))|| − hνE(x, ρ), wi = M ρn−1 Ecc w E(x, ρ) || ¯DχE(Bρ(x))|| ≤ M 2θ η ρ n−1 EccwE(y, η).
Since w∈ νE(y, η)∗ we have EccE(y, η) = EccwE(y, η), hence
2||νE(x, ρ)− νE(y, η)||2s≤ M 2θ η ρ n−1 EccE(y, η)≤M C 2θ η ρ n−1 η2α.
Let now x∈ ∂∗E. Since limρ→0||νE(x, ρ)− νE(x)|| = 0, for ρ < ρ0 we obtain
||νE(x)− νE(x, ρ)||s ≤ ∞ X k=0 ||νE(x, 2−kρ)− νE(x, 2−(k+1)ρ)||s ≤ C0ρα
